proof of the classical limit given on page 54:

’3(n’1)N/2

2πmkB T

n’3nN/2 = (3.78)

h2

n’1

nm

dr 1 · · · dr n’1 exp ’β (r k ’ r k’1 )2 . (3.79)

2β2

2

k=1

It therefore “compensates” in the partition function for the harmonic terms

in the extra degrees of freedom that are introduced by the beads.

Interestingly, the expression for Q in (3.77) is proportional to the partition

function of a system of particles, where each particle i is represented by a

closed string of beads (a “necklace”), with two adjacent beads connected by

a harmonic spring with spring constant

nmi

κi = , (3.80)

2β2

and feeling 1/n of the interparticle potential at the position of each bead.

The interaction V (r i ’ r j ) between two particles i and j acts at the k-th

step between the particles positioned at the k-th node r k . Thus the k-th

node of particle i interacts only with the k-th node of particle j (Fig. 3.5),

with a strength of 1/n times the full interparticle interaction.

The propagator (3.48) used to derive the “string-of-beads” homomor-

phism, is a high-temperature free particle propagator, which “ although in

principle exact “ converges slowly for bound particles in potential wells at low

temperature. Mak and Andersen (1990) have devised a “low-temperature”

propagator that is appropriate for particles in harmonic wells. It contains

the resonance frequency (for example derived from the second derivative of

the potential) and converges faster for bound states with similar frequencies.

3.3 Path integral quantum mechanics 63

16 16

17

gg 14

y

s

8 15

©

¡ 17 15 ££

!9

¡s ©# w

f ¢

t g f ¨¢

£ 14 18

¨¡ t 13

”

18 £ ¨

%i

B

¨7 19 11

££ r

I r 0 13

6 11 k

C

s

d5 10 10

d r

j 0

©

12

¢

12

G

¢

¢

f 19

w A8

4 #

¢I Q

d 2 9 ££

f 3 1

Ay i

4 gg 1 e £

d

‚ …

r

0 3 6r£

j

Q

££ 7

g

)

2 5

Figure 3.5 The paths of two interacting particles. Interactions act between equally-

numbered nodes, with strength 1/n.

For this propagator the Boltzmann term in (3.77) reads

n’1

mω(r k ’ r k’1 )2 2{cosh(β ω/n) ’ 1}

exp ’β + V (r k ) . (3.81)

2 β sinh(β ω/n) β ω sinh(β ω/n)

k=1

The system consisting of strings of beads can be simulated in equilibrium

by conventional classical Monte Carlo (MC) or molecular dynamics (MD)

methods. If MD is used, and the mass of each particle is evenly distributed

over its beads, the time step will become quite small. The oscillation fre-

quency of the bead harmonic oscillators is approximately given by nkB T /h,

which amounts to about 60 THz for a conservative number of ten beads per

necklace, at T = 300 K. Taking 50 steps per oscillation period then requires

a time step as small as 0.3 fs. Such PIMC or PIMD simulations will yield a

set of necklace con¬gurations (one necklace per atom) that is representative

for an equilibrium ensemble at the chosen temperature. The solution is in

principle exact in the limit of an in¬nite number of beads per particle, if

exchange e¬ects can be ignored.

While the PIMC and PIMD simulations are valid for equilibrium systems,

their use in non-equilibrium dynamic simulations is questionable. One may

equilibrate the “quantum part,” i.e., the necklace con¬gurations, at any

given con¬guration of the geometric centers of the necklaces, either by MC

or MD, and compute the necklace-averaged forces between the particles.

Then one may move the system one MD step ahead with those e¬ective

forces. In this way a kind of quantum dynamics is produced, with the mo-

mentum change given by quantum-averaged forces, rather than by forces

evaluated at the quantum-averaged positions. This is exactly what should

be done for the momentum expectation value, according to the derivation

by Ehrenfest (see page 43). One may hope that this method of computing

64 From quantum to classical mechanics: when and how

forces incorporates essential quantum e¬ects in the dynamics of the system.

However, this cannot be proven, as the wave function distribution that is

generated contains no memory of the past dynamics as it should in a full

quantum-dynamical treatment. Neither does this kind of “quantum dynam-

ics” produce a bifurcation into more than one quantum state. Note that

this method cannot handle exchange.4

In Section 3.5 we™ll return to the path-integral methods and employ them

to make approximate quantum corrections to molecular dynamics.

3.4 Quantum hydrodynamics

In this section a di¬erent approach to quantum mechanics is considered,

which originates from Madelung (1926) and de Broglie (1927) in the 1920™s,

and was revived by Bohm (1952a, 1952b) in the ¬fties. It survived the

following decades only in the periphery of the main stream of theoretical

physics, but has more recently regained interest because of its applicabil-

ity to simulation. The approach is characterized by the use of a classical

model consisting of a system of particles or a classical ¬‚uid that “ under